A complete Vinogradov 3-primes theorem under the Riemann hypothesis
Authors:
J.-M. Deshouillers, G. Effinger, H. te Riele and D. Zinoviev
Journal:
Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 99-104
MSC (1991):
Primary 11P32
DOI:
https://doi.org/10.1090/S1079-6762-97-00031-0
Published electronically:
September 17, 1997
MathSciNet review:
1469323
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above $5$ is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation.
- Trudy tret′ego vsesoyuznogo matematicheskogo sЪezda, Moskva, iyun′-iyul′1956. Tom 1. Sektsionnye doklady, Izdat. Akad. Nauk SSSR, Moscow, 1956 (Russian). MR 0100540
- Jing Run Chen and Tian Ze Wang, On the Goldbach problem, Acta Math. Sinica 32 (1989), no. 5, 702–718 (Chinese). MR 1046491
- Jean-Marc Deshouillers, Herman te Riele, and Yannick Saouter, New experimental results concerning the Goldbach conjecture, to appear.
- Gove Effinger, Some numerical implication of the Hardy and Littlewood analysis of the $3$-primes problem, submitted for publication.
- A. Granville, J. van de Lune, and H. J. J. te Riele, Checking the Goldbach conjecture on a vector computer, Number theory and applications (Banff, AB, 1988) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, pp. 423–433. MR 1123087
- G. H. Hardy and L. E. Littlewood, Some problems of ‘Partitio Numerorum’. III: On the expression of a number as a sum of primes, Acta Mathematica 44 (1922), 1–70.
- Aleksandar Ivić, The Riemann zeta-function, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. The theory of the Riemann zeta-function with applications. MR 792089
- Gerhard Jaeschke, On strong pseudoprimes to several bases, Math. Comp. 61 (1993), no. 204, 915–926. MR 1192971, DOI 10.1090/S0025-5718-1993-1192971-8
- Anatolij A. Karatsuba, Basic analytic number theory, Springer-Verlag, Berlin, 1993. Translated from the second (1983) Russian edition and with a preface by Melvyn B. Nathanson. MR 1215269, DOI 10.1007/978-3-642-58018-5
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
- Bruno Lucke, Zur Hardy-Littlewoodschen Behandlung des Goldbachschen Problems, Doctoral Dissertation, Göttingen, 1926.
- Paulo Ribenboim, The book of prime number records, Springer-Verlag, New York, 1988. MR 931080, DOI 10.1007/978-1-4684-9938-4
- J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$, Math. Comp. 29 (1975), 243–269. MR 457373, DOI 10.1090/S0025-5718-1975-0457373-7
- Matti K. Sinisalo, Checking the Goldbach conjecture up to $4\cdot 10^{11}$, Math. Comp. 61 (1993), no. 204, 931–934. MR 1185250, DOI 10.1090/S0025-5718-1993-1185250-6
- I. M. Vinogradov, Representation of an odd number as a sum of three primes, Comptes Rendues (Doklady) de l’Academy des Sciences de l’USSR 15 (1937), 191–294.
- Dmitrii Zinoviev, On Vinogradov’s constant in Goldbach’s ternary problem, Journal of Number Theory, 65 (1997), 334–358.
- K. G. Borodzkin, On I. M. Vinogradov’s constant, Proc. 3rd All-Union Math. Conf., vol. 1, Izdat. Akad. Nauk SSSR, Moscow, 1956. (Russian)
- Jingrun Chen and Tianze Wang, On the odd Goldbach problem, Acta Math. Sinica 32 (1989), 702–718.
- Jean-Marc Deshouillers, Herman te Riele, and Yannick Saouter, New experimental results concerning the Goldbach conjecture, to appear.
- Gove Effinger, Some numerical implication of the Hardy and Littlewood analysis of the $3$-primes problem, submitted for publication.
- A. Granville, J. van de Lune, and H. te Riele, Checking the Goldbach conjecture on a vector computer, Number Theory and Applications (R.A. Mollin, ed.), Kluwer Academic Publishers, 1989, 423–433.
- G. H. Hardy and L. E. Littlewood, Some problems of ‘Partitio Numerorum’. III: On the expression of a number as a sum of primes, Acta Mathematica 44 (1922), 1–70.
- A. Ivic, The Riemann zeta-function, J. Wiley and Sons, 1985.
- Gerhard Jaeschke, On strong pseudoprimes to several bases, Math. Comp. 61 (1993), 915–926.
- A. A. Karatsuba, Basic analytic number theory, Springer-Verlag, 1993.
- U. V. Linnik, The new proof of Goldbach-Vinogradov’s theorem, Mat. Sb. 19 (1946), 3–8.
- Bruno Lucke, Zur Hardy-Littlewoodschen Behandlung des Goldbachschen Problems, Doctoral Dissertation, Göttingen, 1926.
- Paulo Ribenboim, The book of prime number records, Springer-Verlag, 1988.
- Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\Theta (x)$ and $\Psi (x)$, Mathematics of Computation 30 (1976), 337–360.
- Matti K. Sinisalo, Checking the Goldbach conjecture up to $4 \times 10^{11}$, Mathematics of Computation 61 (1993), 931–934.
- I. M. Vinogradov, Representation of an odd number as a sum of three primes, Comptes Rendues (Doklady) de l’Academy des Sciences de l’USSR 15 (1937), 191–294.
- Dmitrii Zinoviev, On Vinogradov’s constant in Goldbach’s ternary problem, Journal of Number Theory, 65 (1997), 334–358.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (1991):
11P32
Retrieve articles in all journals
with MSC (1991):
11P32
Additional Information
J.-M. Deshouillers
Affiliation:
Mathematiques Stochastiques, UMR 9936 CNRS-U.Bordeaux 1, U.Victor Segalen Bordeaux 2, F33076 Bordeaux Cedex, France
Email:
dezou@u-bordeaux2.fr
G. Effinger
Affiliation:
Department of Mathematics and Computer Science, Skidmore College, Saratoga Springs, NY 12866
Email:
effinger@skidmore.edu
H. te Riele
Affiliation:
Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands
Email:
herman.te.riele@cwi.nl
D. Zinoviev
Affiliation:
Memotec Communications, Inc., 600 Rue McCaffrey, Montreal, QC, H4T1N1, Canada
Email:
zinovid@memotec.com
Keywords:
Goldbach,
Vinogradov,
3-primes problem,
Riemann hypothesis
Received by editor(s):
February 26, 1997
Published electronically:
September 17, 1997
Communicated by:
Hugh Montgomery
Article copyright:
© Copyright 1997
American Mathematical Society
